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> Standard complexity theory focuses on answering questions when our substrate behaves like a Turing machine with multiple tapes. This is not true. Complexity

by pron 2mo ago

> Standard complexity theory focuses on answering questions when our substrate behaves like a Turing machine with multiple tapes. This is not true. Complexity theory very much looks at complexity under different models (alphabet size, oracles, circuits). It's just that often (e.g. in the case of alphabets), there is a reduction of known complexity between two models. > It quantifies that distance between prove and verify, but it doesn't study the set of all those distances and how they arise. This is also not true (https://en.wikipedia.org/wiki/Proof_complexity https://en.wikipedia.org/wiki/Proof_complexity).

ux266478 2mo ago
> there is a reduction of known complexity between two models. Every single time I've seen, for example, the lambda calculus be assigned cost semantics, it usually looks like what you would expect out of a Turing machine's simulation of it. Often times, they're explicit about it: https://www.sciencedirect.com/science/article/pii/S0304397508000583 https://www.sciencedirect.com/science/article/pii/S030439750... For me, I can't accept that this is the criteria of "reasonable." Especially not for abstract theory. I did try to indicate I'm mostly talking about standard complexity theory, the stuff you'd encounter on the surface level of the field. I'm not an expert in CT, but I do know enough to know what Landauer's principle is (and that it's been plausibly challenged.) I also know there's some crazy stuff in there, like descriptive complexity theory's link between Existential SOL and NP-Complexity. > This is also not true (https://en.wikipedia.org/wiki/Proof_complexity https://en.wikipedia.org/wiki/Proof_complexity). Do you have any complexity theory papers that deal with this specifically? I've only ever seen that kind of work done in mathematical logic. Genuine interest in reading the CT approach.
- pron 2mo ago
  > it usually looks like what you would expect out of a Turing machine's simulation of it You could assign it any cost model you want. Often this doesn't make a difference (as speedup theorems and other "distracting details" mean that most classes are intended to be separated by exponentials), but it is true that people are typically more interested in cost models that are more relevant to the physical universe (although number of reductions is very much the cost of focus in proof complexity). Indeed, relevant discoveries in physics yield corresponding computational complexity research, as in the case of quantum complexity (https://en.wikipedia.org/wiki/Quantum_complexity_theory https://en.wikipedia.org/wiki/Quantum_complexity_theory). > Do you have any complexity theory papers that deal with this specifically? A Google Scholar search for "proof complexity" will show you many papers as well as a number of books.
  - ux266478 2mo ago
    The point that what gets published adapts to the state of the art is fair. Personally I think its adaptations for ANNs and interaction nets are more interesting than the quantum ones. Once it has to account for the topology of the computer, or deal with non-atomic and continuous substrates, it changes a little bit. But this is orthogonal. The point wasn't that you can't make it adapt, I hope it doesn't come across like I'm dismissing the field and its importance. What I'm trying to say rather specifically is that it's not it's not about studying the nature of computation. It lives at a different level of theory than that. It might just be a disagreement on the article being used. I think it would be fair to say it studies a nature, just not the nature, but that would be true of all fields of CS.
    - pron 2mo ago
      I would say that it very much is about the nature of computation, but computation itself has always been tied to the physical. The physicality (even hypothetical physicality) is exactly what separates theoretical computer science from pure mathematics. Even the two primary resources that complexity measures - time and space - are tied to physical quantities.
      - ux266478 2mo ago
        The distinction is certainly a fair one to make. The etymological root of computation is "done with mental labor", by way of "to clean a financial ledger" (although weirdly, this isn't the real root. Putare is botancial pruning. Computare is a metaphor created after it spent some time on the semantic treadmill.) Grounding it to the physical is perfectly sane, because that's historically how it's been used. That being said, a counterargument to press against this is that complexity theory doesn't restrict itself to physical or hypothetical physicality in its totality. As I mentioned, there are swaths of complexity theory work which bound quite far afield. The higher orders of the field are decidedly not-physical at all (and pedantically, hypercomputation isn't strictly computation). Of course even beyond this, we're still not studying the nature of computation, we're studying the cost. While you might say that of course these things are tied thanks to physicality (I wouldn't agree that they're equivalent on this basis, but I don't think that's an interesting semantic argument), I did also mention Landauer's principle being plausibly challenged, which is further problematic for conflating the two. Computation being reversible where entropy isn't doesn't explode complexity theory, but it does drive a wedge between information in a computer-sense and information in a thermodynamics sense. At that point, we don't have the claim in the first place, it's just a false friend. Something to consider.

by pron 2mo ago

ux266478 2mo ago

> there is a reduction of known complexity between two models. Every single time I've seen, for example, the lambda calculus be assigned cost semantics, it usually looks like what you would expect out of a Turing machine's simulation of it. Often times, they're explicit about it: https://www.sciencedirect.com/science/article/pii/S0304397508000583 https://www.sciencedirect.com/science/article/pii/S030439750... For me, I can't accept that this is the criteria of "reasonable." Especially not for abstract theory. I did try to indicate I'm mostly talking about standard complexity theory, the stuff you'd encounter on the surface level of the field. I'm not an expert in CT, but I do know enough to know what Landauer's principle is (and that it's been plausibly challenged.) I also know there's some crazy stuff in there, like descriptive complexity theory's link between Existential SOL and NP-Complexity. > This is also not true (https://en.wikipedia.org/wiki/Proof_complexity https://en.wikipedia.org/wiki/Proof_complexity). Do you have any complexity theory papers that deal with this specifically? I've only ever seen that kind of work done in mathematical logic. Genuine interest in reading the CT approach.

pron 2mo ago
> it usually looks like what you would expect out of a Turing machine's simulation of it You could assign it any cost model you want. Often this doesn't make a difference (as speedup theorems and other "distracting details" mean that most classes are intended to be separated by exponentials), but it is true that people are typically more interested in cost models that are more relevant to the physical universe (although number of reductions is very much the cost of focus in proof complexity). Indeed, relevant discoveries in physics yield corresponding computational complexity research, as in the case of quantum complexity (https://en.wikipedia.org/wiki/Quantum_complexity_theory https://en.wikipedia.org/wiki/Quantum_complexity_theory). > Do you have any complexity theory papers that deal with this specifically? A Google Scholar search for "proof complexity" will show you many papers as well as a number of books.
- ux266478 2mo ago
  The point that what gets published adapts to the state of the art is fair. Personally I think its adaptations for ANNs and interaction nets are more interesting than the quantum ones. Once it has to account for the topology of the computer, or deal with non-atomic and continuous substrates, it changes a little bit. But this is orthogonal. The point wasn't that you can't make it adapt, I hope it doesn't come across like I'm dismissing the field and its importance. What I'm trying to say rather specifically is that it's not it's not about studying the nature of computation. It lives at a different level of theory than that. It might just be a disagreement on the article being used. I think it would be fair to say it studies a nature, just not the nature, but that would be true of all fields of CS.
  - pron 2mo ago
    I would say that it very much is about the nature of computation, but computation itself has always been tied to the physical. The physicality (even hypothetical physicality) is exactly what separates theoretical computer science from pure mathematics. Even the two primary resources that complexity measures - time and space - are tied to physical quantities.
    - ux266478 2mo ago
      The distinction is certainly a fair one to make. The etymological root of computation is "done with mental labor", by way of "to clean a financial ledger" (although weirdly, this isn't the real root. Putare is botancial pruning. Computare is a metaphor created after it spent some time on the semantic treadmill.) Grounding it to the physical is perfectly sane, because that's historically how it's been used. That being said, a counterargument to press against this is that complexity theory doesn't restrict itself to physical or hypothetical physicality in its totality. As I mentioned, there are swaths of complexity theory work which bound quite far afield. The higher orders of the field are decidedly not-physical at all (and pedantically, hypercomputation isn't strictly computation). Of course even beyond this, we're still not studying the nature of computation, we're studying the cost. While you might say that of course these things are tied thanks to physicality (I wouldn't agree that they're equivalent on this basis, but I don't think that's an interesting semantic argument), I did also mention Landauer's principle being plausibly challenged, which is further problematic for conflating the two. Computation being reversible where entropy isn't doesn't explode complexity theory, but it does drive a wedge between information in a computer-sense and information in a thermodynamics sense. At that point, we don't have the claim in the first place, it's just a false friend. Something to consider.